Are $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| x^4 , |y| < 10 right} $ homeomorphic?
up vote
0
down vote
favorite
Are spaces $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| < 10 right} $ and $left{(x,y) in mathbb{R^2} | |y| > x^4 , |y| < 10 right} $ homeomorphic? I think they might be but can't construct a homeomorphism because one is a subset of the other.
general-topology continuity
asked Nov 18 at 14:31
user15269
1608
add a comment |
up vote
0
down vote
favorite
Are spaces $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| < 10 right} $ and $left{(x,y) in mathbb{R^2} | |y| > x^4 , |y| < 10 right} $ homeomorphic? I think they might be but can't construct a homeomorphism because one is a subset of the other.
general-topology continuity
asked Nov 18 at 14:31
user15269
1608
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Are spaces $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| < 10 right} $ and $left{(x,y) in mathbb{R^2} | |y| > x^4 , |y| < 10 right} $ homeomorphic? I think they might be but can't construct a homeomorphism because one is a subset of the other.
general-topology continuity
asked Nov 18 at 14:31
user15269
1608
Are spaces $left{ (x,y) in mathbb{R^2} | |y| > x^2 , |y| < 10 right} $ and $left{(x,y) in mathbb{R^2} | |y| > x^4 , |y| < 10 right} $ homeomorphic? I think they might be but can't construct a homeomorphism because one is a subset of the other.
general-topology continuity
general-topology continuity
asked Nov 18 at 14:31
user15269
1608
asked Nov 18 at 14:31
user15269
1608
edited Nov 19 at 9:15
asked Nov 18 at 14:31
user15269
1608
asked Nov 18 at 14:31
user15269
1608
asked Nov 18 at 14:31
user15269
1608
1608
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19
add a comment |
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19
sorry I corrected it
– user15269
Nov 19 at 9:19
add a comment |
1 Answer
1
active
oldest
votes
up vote
1
down vote
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
10.6k1527
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
10.6k1527
add a comment |
up vote
1
down vote
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
10.6k1527
add a comment |
up vote
1
down vote
up vote
1
down vote
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
10.6k1527
Let's disect those sets. Both of them consist of two connected components: when $y<0$ and when $y>0$. It is enough that we show that those components are pairwise homeomorphic. And since each upper half component is a reflection of the lower half then it is enough to show that $A={(x,y) | y>x^2; y< 10}$ and $B={(x,y) | y>x^4; y< 10}$ are homeomorphic.
To do that you have to realize that both $f(x)=x^2$ and $f(x)=x^4$ are convex functions. A bit of work has to be done to make sure that these conditions together with $y<10$ imply that both $A$ and $B$ are open and convex subsets of $mathbb{R}^2$. And it is well known that every open and convex subset of $mathbb{R}^n$ is homeomorphic to a $n$-ball.
answered Nov 19 at 9:29
freakish
10.6k1527
answered Nov 19 at 9:29
freakish
10.6k1527
answered Nov 19 at 9:29
freakish
10.6k1527
answered Nov 19 at 9:29
freakish
10.6k1527
10.6k1527
add a comment |
add a comment |
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3003604%2fare-left-x-y-in-mathbbr2-y-x2-y-10-right-left%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Can you construct a homeomorphism between e.g. ${(x,y)mid x^2+y^2leq1}$ and ${(x,y)mid x^2+y^2leq4}$? Also here one is a subset of the other. I don't see why that should be an obstacle.
– drhab
Nov 18 at 14:46
What is (x,y)(x,y)?
– William Elliot
Nov 19 at 2:41
sorry I corrected it
– user15269
Nov 19 at 9:19